# Proof or reference to the Weyl inequalities?

Does anyone know a proof or reference to the following result?

Suppose that $A, B$ are both $m \times n$ real matrices. Then for all $1 \leq k \leq \min\{m, n\}$, $$|\sigma_k(A) - \sigma_k(B)| \leq \|A - B\|.$$

I think these are called the Weyl inequalities, and I remember learning a proof of this result using the minimax characterization of these singular values but I can't reconstruct the proof. Anyone know it or have a reference to it?

• @QiaochuYuan Beautiful set of notes, by the way. You should make a comment and I could select it as the selected answer? May 16, 2018 at 8:19

$$\sigma_{k+\ell+1}(A + B) \le \sigma_{k+1}(A) + \sigma_{\ell+1}(B)$$